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Introductory Business Statistics

B | Mathematical Phrases, Symbols, and Formulas

Introductory Business StatisticsB | Mathematical Phrases, Symbols, and Formulas
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  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. Key Terms
    7. Chapter Review
    8. Homework
    9. References
    10. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Display Data
    3. 2.2 Measures of the Location of the Data
    4. 2.3 Measures of the Center of the Data
    5. 2.4 Sigma Notation and Calculating the Arithmetic Mean
    6. 2.5 Geometric Mean
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. Key Terms
    10. Chapter Review
    11. Formula Review
    12. Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables and Probability Trees
    6. 3.5 Venn Diagrams
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Bringing It Together: Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Hypergeometric Distribution
    3. 4.2 Binomial Distribution
    4. 4.3 Geometric Distribution
    5. 4.4 Poisson Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Properties of Continuous Probability Density Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Estimating the Binomial with the Normal Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means
    3. 7.2 Using the Central Limit Theorem
    4. 7.3 The Central Limit Theorem for Proportions
    5. 7.4 Finite Population Correction Factor
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Confidence Interval for a Population Standard Deviation, Known or Large Sample Size
    3. 8.2 A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case
    4. 8.3 A Confidence Interval for A Population Proportion
    5. 8.4 Calculating the Sample Size n: Continuous and Binary Random Variables
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Full Hypothesis Test Examples
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Comparing Two Independent Population Means
    3. 10.2 Cohen's Standards for Small, Medium, and Large Effect Sizes
    4. 10.3 Test for Differences in Means: Assuming Equal Population Variances
    5. 10.4 Comparing Two Independent Population Proportions
    6. 10.5 Two Population Means with Known Standard Deviations
    7. 10.6 Matched or Paired Samples
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Test of a Single Variance
    4. 11.3 Goodness-of-Fit Test
    5. 11.4 Test of Independence
    6. 11.5 Test for Homogeneity
    7. 11.6 Comparison of the Chi-Square Tests
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  13. 12 F Distribution and One-Way ANOVA
    1. Introduction
    2. 12.1 Test of Two Variances
    3. 12.2 One-Way ANOVA
    4. 12.3 The F Distribution and the F-Ratio
    5. 12.4 Facts About the F Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  14. 13 Linear Regression and Correlation
    1. Introduction
    2. 13.1 The Correlation Coefficient r
    3. 13.2 Testing the Significance of the Correlation Coefficient
    4. 13.3 Linear Equations
    5. 13.4 The Regression Equation
    6. 13.5 Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation
    7. 13.6 Predicting with a Regression Equation
    8. 13.7 How to Use Microsoft Excel® for Regression Analysis
    9. Key Terms
    10. Chapter Review
    11. Practice
    12. Solutions
  15. A | Statistical Tables
  16. B | Mathematical Phrases, Symbols, and Formulas
  17. Index

English Phrases Written Mathematically

When the English says: Interpret this as:
X is at least 4. X ≥ 4
The minimum of X is 4. X ≥ 4
X is no less than 4. X ≥ 4
X is greater than or equal to 4. X ≥ 4
X is at most 4. X ≤ 4
The maximum of X is 4. X ≤ 4
X is no more than 4. X ≤ 4
X is less than or equal to 4. X ≤ 4
X does not exceed 4. X ≤ 4
X is greater than 4. X > 4
X is more than 4. X > 4
X exceeds 4. X > 4
X is less than 4. X < 4
There are fewer X than 4. X < 4
X is 4. X = 4
X is equal to 4. X = 4
X is the same as 4. X = 4
X is not 4. X ≠ 4
X is not equal to 4. X ≠ 4
X is not the same as 4. X ≠ 4
X is different than 4. X ≠ 4
Table B1

Symbols and Their Meanings

Chapter (1st used) Symbol Spoken Meaning
Sampling and Data                     The square root of same
Sampling and Data π π Pi 3.14159… (a specific number)
Descriptive Statistics Q1 Quartile one the first quartile
Descriptive Statistics Q2 Quartile two the second quartile
Descriptive Statistics Q3 Quartile three the third quartile
Descriptive Statistics IQR interquartile range Q3Q1 = IQR
Descriptive Statistics x x x-bar sample mean
Descriptive Statistics μμ mu population mean
Descriptive Statistics s s sample standard deviation
Descriptive Statistics s2s2 s squared sample variance
Descriptive Statistics σσ sigma population standard deviation
Descriptive Statistics σ2σ2 sigma squared population variance
Descriptive Statistics ΣΣ capital sigma sum
Probability Topics { } {} brackets set notation
Probability Topics SS S sample space
Probability Topics AA Event A event A
Probability Topics P ( A ) P ( A ) probability of A probability of A occurring
Probability Topics P(A|B) P(A|B) probability of A given B prob. of A occurring given B has occurred
Probability Topics P(AB) P(AB) prob. of A or B prob. of A or B or both occurring
Probability Topics P(AB) P(AB) prob. of A and B prob. of both A and B occurring (same time)
Probability Topics A A-prime, complement of A complement of A, not A
Probability Topics P(A') prob. of complement of A same
Probability Topics G1 green on first pick same
Probability Topics P(G1) prob. of green on first pick same
Discrete Random Variables PDF prob. density function same
Discrete Random Variables X X the random variable X
Discrete Random Variables X ~ the distribution of X same
Discrete Random Variables greater than or equal to same
Discrete Random Variables less than or equal to same
Discrete Random Variables = equal to same
Discrete Random Variables not equal to same
Continuous Random Variables f(x) f of x function of x
Continuous Random Variables pdf prob. density function same
Continuous Random Variables U uniform distribution same
Continuous Random Variables Exp exponential distribution same
Continuous Random Variables f(x) = f of x equals same
Continuous Random Variables m m decay rate (for exp. dist.)
The Normal Distribution N normal distribution same
The Normal Distribution z z-score same
The Normal Distribution Z standard normal dist. same
The Central Limit Theorem X X X-bar the random variable X-bar
The Central Limit Theorem μ x μ x mean of X-bars the average of X-bars
The Central Limit Theorem σ x σ x standard deviation of X-bars same
Confidence Intervals CL confidence level same
Confidence Intervals CI confidence interval same
Confidence Intervals EBM error bound for a mean same
Confidence Intervals EBP error bound for a proportion same
Confidence Intervals t Student's t-distribution same
Confidence Intervals df degrees of freedom same
Confidence Intervals t α 2 t α 2 student t with α/2 area in right tail same
Confidence Intervals p p p-prime sample proportion of success
Confidence Intervals q q q-prime sample proportion of failure
Hypothesis Testing H 0 H 0 H-naught, H-sub 0 null hypothesis
Hypothesis Testing H a H a H-a, H-sub a alternate hypothesis
Hypothesis Testing H 1 H 1 H-1, H-sub 1 alternate hypothesis
Hypothesis Testing αα alpha probability of Type I error
Hypothesis Testing β β beta probability of Type II error
Hypothesis Testing X1 X2 ¯ X1 X2 ¯ X1-bar minus X2-bar difference in sample means
Hypothesis Testing μ 1 μ 2 μ 1 μ 2 mu-1 minus mu-2 difference in population means
Hypothesis Testing P 1 P 2 P 1 P 2 P1-prime minus P2-prime difference in sample proportions
Hypothesis Testing p 1 p 2 p 1 p 2 p1 minus p2 difference in population proportions
Chi-Square Distribution Χ 2 Χ 2 Ky-square Chi-square
Chi-Square Distribution OO Observed Observed frequency
Chi-Square Distribution EE Expected Expected frequency
Linear Regression and Correlation y = a + bx y equals a plus b-x equation of a straight line
Linear Regression and Correlation y^y^ y-hat estimated value of y
Linear Regression and Correlation rr sample correlation coefficient same
Linear Regression and Correlation εε error term for a regression line same
Linear Regression and Correlation SSE Sum of Squared Errors same
F-Distribution and ANOVA F F-ratio F-ratio
Table B2 Symbols and their Meanings

Formulas

Symbols you must know
Population Sample
NN Size nn
μμ Mean x_x_
σ2σ2 Variance s2s2
σσ Standard deviation ss
pp Proportion pp
Single data set formulae
Population Sample
μ = E(x) = 1N i=1 N (xi) μ=E(x)=1N i=1 N (xi) Arithmetic mean x = 1n i=1 n (xi) x =1n i=1 n (xi)
Geometric mean x~ = (i=1nXi) 1nx~= (i=1nXi) 1n
Q3=3(n+1)4Q3=3(n+1)4, Q1=(n+1)4Q1=(n+1)4 Inter-quartile range
IQR=Q3Q1IQR=Q3Q1
Q3=3(n+1)4Q3=3(n+1)4, Q1=(n+1)4Q1=(n+1)4
σ2=1N i=1 N (xiμ)2σ2=1N i=1 N (xiμ)2 Variance s2=1n i=1 n (xix_)2s2=1n i=1 n (xix_)2
Single data set formulae
Population Sample
μ = E(x) = 1N i=1 N (mi·fi) μ=E(x)=1N i=1 N (mi·fi) Arithmetic mean x = 1n i=1 n (mi·fi) x =1n i=1 n (mi·fi)
Geometric mean x~ = (i=1nXi) 1nx~= (i=1nXi) 1n
σ2=1N i=1 N (miμ)2 ·fiσ2=1N i=1 N (miμ)2·fi Variance s2=1n i=1 n (mix_)2·fis2=1n i=1 n (mix_)2·fi
CV=σμ·100CV=σμ·100 Coefficient of variation CV=sx_·100CV=sx_·100
Table B3
Basic probability rules
P(AB)=P(A|B)·P(B)P(AB)=P(A|B)·P(B) Multiplication rule
P(AB)=P(A)+P(B)P(AB)P(AB)=P(A)+P(B)P(AB) Addition rule
P(AB)=P(A)·P(B)P(AB)=P(A)·P(B) or P(A|B)=P(A)P(A|B)=P(A) Independence test
Hypergeometric distribution formulae
nCx=(nx)=n!x!(nx)!nCx=(nx)=n!x!(nx)! Combinatorial equation
P(x)=(Ax)(NAnx)(Nn)P(x)=(Ax)(NAnx)(Nn) Probability equation
E(X)=μ=npE(X)=μ=np Mean
σ2=(NnN1)np(q)σ2=(NnN1)np(q) Variance
Binomial distribution formulae
P(x)=n!x!(nx)!px(q)nxP(x)=n!x!(nx)!px(q)nx Probability density function
E(X)=μ=npE(X)=μ=np Arithmetic mean
σ2=np(q)σ2=np(q) Variance
Geometric distribution formulae
P(X=x)=(1p)x1(p)P(X=x)=(1p)x1(p) Probability when xx is the first success. Probability when xx is the number of failures before first success P(X=x)=(1p)x(p)P(X=x)=(1p)x(p)
μ=1pμ=1p Mean Mean μ=1ppμ=1pp
σ2=(1p)p2σ2=(1p)p2 Variance Variance σ2=(1p)p2σ2=(1p)p2
Poisson distribution formulae
P(x)=eμμxx!P(x)=eμμxx! Probability equation
E(X)=μE(X)=μ Mean
σ2=μσ2=μ Variance
Uniform distribution formulae
f(x)=1baf(x)=1ba for axbaxb PDF
E(X)=μ=a+b2E(X)=μ=a+b2 Mean
σ2=(ba)212σ2=(ba)212 Variance
Exponential distribution formulae
P(Xx)=1emxP(Xx)=1emx Cumulative probability
E(X)=μ=1mE(X)=μ=1m or m=1μm=1μ Mean and decay factor
σ2=1m2=μ2σ2=1m2=μ2 Variance
Table B4
The following page of formulae requires the use of the "ZZ", "tt", "χ2χ2" or "FF" tables.
Z=xμσZ=xμσ Z-transformation for normal distribution
Z=xnpnp(q)Z=xnpnp(q) Normal approximation to the binomial
Probability (ignores subscripts)
Hypothesis testing
Confidence intervals
[bracketed symbols equal margin of error]
(subscripts denote locations on respective distribution tables)
Zc=x-μ0σnZc=x-μ0σn Interval for the population mean when sigma is known
x±[Z(α/2)σn]x±[Z(α/2)σn]
Zc=x-μ0snZc=x-μ0sn Interval for the population mean when sigma is unknown but n>30n>30
x±[Z(α/2)sn]x±[Z(α/2)sn]
tc=x-μ0sntc=x-μ0sn Interval for the population mean when sigma is unknown but n<30n<30
x±[t(n1),(α/2)sn]x±[t(n1),(α/2)sn]
Zc=p-p0p0q0nZc=p-p0p0q0n Interval for the population proportion
p±[Z(α/2)pqn]p±[Z(α/2)pqn]
tc=d-δ0sdntc=d-δ0sdn Interval for difference between two means with matched pairs
d±[t(n1),(α/2)sdn]d±[t(n1),(α/2)sdn] where sdsd is the deviation of the differences
Zc=(x1-x2)δ0σ12n1+σ22n2Zc=(x1-x2)δ0σ12n1+σ22n2 Interval for difference between two means when sigmas are known
(x1-x2)±[Z(α/2)σ12n1+σ22n2](x1-x2)±[Z(α/2)σ12n1+σ22n2]
tc=(x¯1-x¯2)-δ0((s1)2n1+(s2)2n2)tc=(x¯1-x¯2)-δ0((s1)2n1+(s2)2n2) Interval for difference between two means with equal variances when sigmas are unknown
(x¯1-x¯2)±[tdf,(α/2)((s1)2n1+(s2)2n2)](x¯1-x¯2)±[tdf,(α/2)((s1)2n1+(s2)2n2)] where df= ( (s1)2n1 + (s2)2n2 )2 (1n11) ((s1)2n1) + (1n21) ((s2)2n2) df= ( (s1)2n1 + (s2)2n2 )2 (1n11) ((s1)2n1) + (1n21) ((s2)2n2)
Zc=(p1-p2)δ0p1(q1)n1+p2(q2)n2Zc=(p1-p2)δ0p1(q1)n1+p2(q2)n2 Interval for difference between two population proportions
(p1-p2)±[Z(α/2)p1(q1)n1+p2(q2)n2 ](p1-p2)±[Z(α/2)p1(q1)n1+p2(q2)n2]
χc2=(n1)s2σ02χc2=(n1)s2σ02 Tests for GOF, Independence, and Homogeneity
χc2=Σ(OE)2Eχc2=Σ(OE)2Ewhere O = observed values and E = expected values
Fc=s12s22Fc=s12s22 Where s12s12 is the sample variance which is the larger of the two sample variances
The next 3 formulae are for determining sample size with confidence intervals.
(note: E represents the margin of error)
n=Z(a2)2σ2E2n=Z(a2)2σ2E2
Use when sigma is known
E=x¯μE=x¯μ
n=Z(a2)2(0.25)E2n=Z(a2)2(0.25)E2
Use when pp is unknown
E=ppE=pp
n=Z(a2)2[p(q)]E2n=Z(a2)2[p(q)]E2
Use when pp is uknown
E=ppE=pp
Table B5
Simple linear regression formulae for y=a+b(x)y=a+b(x)
r=Σ[(xx¯)(yy¯)]Σ(xx¯)2*Σ(yy¯)2=SxySxSy=SSRSSTr=Σ[(xx¯)(yy¯)]Σ(xx¯)2*Σ(yy¯)2=SxySxSy=SSRSST Correlation coefficient
b=Σ[(xx¯)(yy¯)]Σ(xx¯)2=SxySSx=ry,x(sysx)b=Σ[(xx¯)(yy¯)]Σ(xx¯)2=SxySSx=ry,x(sysx) Coefficient b (slope)
a=y¯b(x¯)a=y¯b(x¯) y-intercept
se2=Σ(yiy^i)2 nk=Σi=1nei2nkse2=Σ(yiy^i)2 nk=Σi=1nei2nk Estimate of the error variance
Sb=se2(xix¯)2=se2(n1)sx2Sb=se2(xix¯)2=se2(n1)sx2 Standard error for coefficient b
tc=bβ0sbtc=bβ0sb Hypothesis test for coefficient β
b±[tn2,α/2Sb]b±[tn2,α/2Sb] Interval for coefficient β
y^±[tα/2*se(1n+(xpx¯)2sx)]y^±[tα/2*se(1n+(xpx¯)2sx)] Interval for expected value of y
y^±[tα/2*se(1+1n+(xpx¯)2sx)]y^±[tα/2*se(1+1n+(xpx¯)2sx)] Prediction interval for an individual y
ANOVA formulae
SSR=Σi=1n(y^iy¯)2SSR=Σi=1n(y^iy¯)2 Sum of squares regression
SSE=Σi=1n(y^iy¯i)2SSE=Σi=1n(y^iy¯i)2 Sum of squares error
SST=Σi=1n(yiy¯)2SST=Σi=1n(yiy¯)2 Sum of squares total
R2=SSRSSTR2=SSRSST Coefficient of determination
Table B6
The following is the breakdown of a one-way ANOVA table for linear regression.
Source of variation Sum of squares Degrees of freedom Mean squares F-ratio
Regression SSRSSR 11 or k1k1 MSR=SSRdfRMSR=SSRdfR F=MSRMSEF=MSRMSE
Error SSESSE nknk MSE=SSEdfEMSE=SSEdfE
Total SSTSST n1n1
Table B7
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